Polyphase Isomorphism between Modular Arithmetic and Multirate Signal Processing

Polyphase Isomorphism between Modular Arithmetic and Multirate Signal Processing "An exact mathematical bridge between the arithmetic of integers and the filter banks of digital signal processing." 🎯 TL;DR — The Essentials What is this? A formal proof that modular decomposition of integer-indexed series is isomorphic to polyphase decomposition in DSP. The Theorem: The projection of a series ∑an onto congruence classes modulo m is mathematically identical to the decimation of the signal xn=an by factor M=m. The Consequence: Techniques from number theory (Euler products, modular representations of constants) transfer directly to filter bank design, and tools from DSP (perfect reconstruction, noble identities, multirate identities) apply to the evaluation of infinite series. The Validation: A fully reproducible Colab notebook illustrating the isomorphism on the Leibniz series for π, the modular Euler product, and the perfect reconstruction property. The Result: A peer-reviewed paper and an executable notebook bridging two disciplines that have developed independently for decades. 🌌 Overview This project presents the theoretical framework and computational validation of the article "Polyphase Isomorphism between Modular Arithmetic and Multirate Signal Processing". Two classical disciplines—the theory of numbers and digital signal processing—deal with sequences indexed by integers. In number theory, one studies infinite series ∑an and decomposes them by congruence classes modulo m. In DSP, one decomposes discrete-time signals into polyphase components by decimating with factor M. This work proves that these two operations are formally identical. 🧩 The Thesis: One Isomorphism, Two Domains We establish that the modular projection Sr=∑kamk+r and the polyphase component Er(z)=∑kamk+rz−k evaluated at z=1 are exact mathematical counterparts. The isomorphism guarantees perfect reconstruction in both domains and enables the transfer of tools across disciplines. 🚀 Key Scientific Contributions 1. The Polyphase Isomorphism Theorem (Section 3) We formally prove that, for any absolutely convergent series an and any modulus m≥2, the modular decomposition into residue classes and the polyphase decomposition with decimation factor M=m are isomorphic: Sr=Er(1),r=0,…,m−1 The orthogonality of the channels in the index domain guarantees perfect reconstruction without aliasing. 2. The Hilbert Space of Prime Residues (Section 2) For the special case m=6, the resonant channels (r=1,5) form a Hilbert space V6 spanned by the orthonormal basis |1⟩,|5⟩. The Leibniz series for π is interpreted as a projection onto this space, revealing its internal arithmetic structure. 3. Applications in Number Theory (Section 4) Modular Leibniz series: π=3∑k(−1)k(16k+1+16k+5) Modular Euler product: π=9⋅∏p>3, p≡±1(mod6)p2p2−1 The isomorphism applies to any series with predictable modular structure. 4. Applications in Signal Processing (Section 5) Perfect reconstruction: The disjointness of modular channels guarantees alias-free filter banks. Shared-Nothing architecture: Each polyphase component can be processed independently, enabling embarrassingly parallel computation. Complexity reduction: Decimation by m reduces recursion tree depth by log2⁡m. 📊 Visual Validation: Perfect Reconstruction The Colab notebook verifies the isomorphism numerically on the Leibniz series. The modular channels sum exactly to the original series value, confirming the perfect reconstruction property to machine precision. Figure 1. Convergence comparison between the direct Leibniz series and its modular decomposition (m=6). Both converge to the same limit, but the modular version reveals the internal arithmetic structure. 🧩 Structural Unification: Reformulating Classical Series The modular paradigm applies not only to π but to any constant defined by an integer-indexed series, revealing hidden symmetries and enabling new parallelization strategies. Concept Classical Formula Modular Reformulation Leibniz π4=∑k=0∞(−1)k2k+1 π=3∑(−1)k(16k+1+16k+5) Catalan G=∑k=0∞(−1)k(2k+1)2 Decomposition over C1⊕C5 Apéry ζ(3)=∑n=1∞1n3 Separate sums over prime channels Hypergeometric ∑(a)n(b)n(c)nn!zn Polyphase decimation by m 💡 Computational Implication: The decomposition allows decoupling the calculation into m independent threads (one per modular channel) with zero data dependencies—ideal for GPU/FPGA implementations. 💻 Reproducibility All code has been designed to be fully auditable and reproducible. Requirements: Python 3.10+ numpy, matplotlib, sympy Cloud Execution: You can replicate the modular decomposition, perfect reconstruction verification, and Euler product convergence directly in Google Colab: 🌐 The Broader Research Programme This work is part of a larger investigation into the physical and computational consequences of the Z/6Z modular symmetry of the Standard Model vacuum. Related projects include: Modular-Substrate-Theory: Unified framework for cosmology and hadronic physics. Arquitectura-de-Hibridacion-Algoritmica-en-Z-6Z: DSP architecture for extreme-precision π computation. Phase-Pi-Quantum-Prior: Topological state preparation and dissipative protection for quantum registers. Common Thread: All projects leverage modular arithmetic (Z/6Z) as a fundamental organising principle across mathematics, physics, and computation. ❤️ Support Independent Science This work is the result of independent research, without institutional funding. The authority of science lies in evidence, not affiliation. If you value this effort: ⭐ Star Github repository. 📢 Share the findings on Twitter/LinkedIn. 💬 Open an Issue if you have ideas to extend the theory. Author: José Ignacio Peinador Sala Contact: joseignacio.peinador@gmail.com